#8508877

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(2,5)(6,8)(9,10)(11,20)(12,18)(13,17)(14,16)(15,19)(21,23)(22,24)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(3,7)(6,8)(9,11,15,14)(10,16,19,20)(12,21,13,22)(17,23,18,24)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,17,19,18)(11,21,14,22)(16,23,20,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(1,3,4,7)(2,6,5,8)(9,13,15,12)(10,18,19,17)(11,21,14,22)(16,23,20,24)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,24)(22,23)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40);(9,14,15,11)(10,16,19,20)(12,22,13,21)(17,23,18,24)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416073);